Birthday Probabilities

The paradox of birthdays is a mathematical problem put forward by Von Mises, who looks for the value N in the problem: In a group of N people there is 50% chance that at least 2 people in the group share the same birthday (day + month). The answer is N = 23, which is quite counterintuitive, hence the paradox.

During the calculation of the birthdate paradox, it is supposed that births are equally distributed over the days of a year (it is not exactly true in reality). In the following, a year has 365 days (leap years are ignored).

Probability is 1/365 = 0.0027 = 0.27%, indeed, one chance out of 365 to be born on a precise day.

The probability is 364/365 = 0.9973 = 99.73%, indeed, he must be born on a distinct day of mine, so there is 364 possibilities out of 365.

This calculation is the same as What is the probability for a person to be born on a given day of the year? This given day is my birthday.

The calculation could also be formulated by noting that the probability for a person to be born the same day as me is the opposite as the probability for a person to be born on a different day than mine.

It is also the probability that a mother has a second child born on the same day (not the same year) as the first.

This calculation is the same as What is the probability for a person (the child) to be born on a given day of the year (the mother birthday)?.

The probability that a mother and her 2 children were born on the same day

The probability that a mother and her N children were born on the same day is 1 in 365 ^ N

$$ P(X) = 1/365^N $$

For N = 2, it is the same as probability for a person to be born the same day as me or the opposite as the probability for a person to be born on a different day than mine.

For other people, they have to be born also a distinct day than me, but also a distinct day of the other one (then calculate the opposite):

A group of 23 people has a probability of approximately 0.5, 50% (1 chance out of 2) that 2 people have a common birth date (day + month).

366 are needed (367 if leap years are taken into account). Indeed, every day of the year plus one are needed to be sure that at least a couple of two people share the same birthday.

For a given date, 253 people are needed to get a probability of 50% that 2 people to be born on a same precise day.

Probabilities are multiplied

The probability that all N people are not born on a given date is P = (364/365)^N

With the hypothesis of a world population of 8 billion people, there are 8000000000 / 365.25 or about 22 million people born on a given day / month.

Limiting to the USA (350 millions inhabitants), there are 350000000 / 365.25 or nearly 1 millions US people born on a given day / month date.